### A 3-adic Faltings-Serre Method

Linfoot Number Theory Seminar

15th January 2020, 11:00 am – 12:00 pm

Fry Building, 2.04

Determining Galois representations and proving isomorphisms between them have a crucial role in modern number theory. One of the strongest tools we have for this is the Faltings-Serre-Livné method for two dimensional Galois representations that take values in a finite extension of Q_2, and there has been extensive effort to convert these theoretical results into a deterministic and implementable algorithm. Ideally we would like to have an effective Faltings-Serre that works for a general n-dimensional Galois representation with values in a general local field. In this seminar I will discuss the main ideas and results that lead to an effective Faltings-Serre method for two dimensional Galois representations with values in Q_3, and time permitting, its connection with the most recent result on modularity lifting due to Allen et al.2019. The implementation is joint work with John Cremona.

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